Am I right in thinking you are doing some laser heating here? if so you should look at
the paper written by Bechtel "heating of solid targets with laser pulses" . Also check out the citations there are a lot of papers out there that solve this kind of problem

This is a transient conduction problem that can be made much easier by assuming you only need to know the temperature about a second after the heat flux is turned off, not before. The reason is that the characteristic diffusion time through 1 mm of steel is about a tenth of a second:

where D is the thermal diffusivity and h is the thickness, and I've used your numbers. After about a second, the temperature will have become nearly uniform in the plate, and the temperature increase can be calculated by considering the amount of input energy: 300 J/m2.

[tex]\Delta T=\frac{E}{c\rho h}[/tex]

I get about a tenth of a degree Celsius for the temperature increase. Does this help?